OK, let's look at it this way.
The mom & dad have raised their son to completely eschew cursing, and to recognize it as offensive behavior when done in public.
Facebook *is* public. Everybody sees it; it gets pushed out to them. It's the equivalent of swearing on the street--a street populated by people who actually *know* who you are. So, what you do is more closely linked to your reputation there than it would be if you swore on the street in a completely strange city. It matters *more* when it's among people you know.
The soldier may be in Afghanistan, but he is *on* Facebook, and that's "the public street in your hometown."
The fact that he used a vulgarity that he apparently was strongly taught not to use means that he is redefining what "proper behavior in public" is, or that he is not recognizing the concept that "Facebook is 'in public', or that he is so stressed by his assignment that he is unable to exercise the self-control that he has used all his life.
And his father is rightfully concerned about this. Even when we are grownups, our parents have a stake in having "raised us right." Our parents have the right to say, "this is dishonorable behavior; our family doesn't behave like this," whether it's cheating grandma out of money, abandoning a pregnant girlfriend, all the way down to using swear words in front of people you *know* will find them offensive, or belching at the dinner table and not saying "excuse me."
However, *publicly* chiding their son is not good. Not even if they are worried that people will say, "Johnny used the F-word on his Facebook status? I thought Marge and Ralph raised him better than that!"
He should have sent a message to his son's Facebook in-box and said, "I'm concerned--that was sort of shocking, esp. for you, knowing what we tried to teach you. What's going on? Oh, and knock it off, if you would, please."